00:01
In this case we got the set of two by two matrices of this form that are basically diagonal matrices.
00:11
And the sum and scalar multiplication is defined the usual way.
00:15
That means if i pick two elements of this set m, and i summed them up, i obtained this, just some element by element of the matrices.
00:28
And we obtain this.
00:31
And from here it's important because here we can check that what we obtain is again a diagonal matrix, that means that the operation is closed.
00:39
So this part here actually prove the first action of vector spaces.
00:47
And the square multiplication is defined in the same manner, it's just multiplication by each of the elements of the matrix.
00:55
And we obtained again a diagonal matrix.
00:57
Again, this is still on the set m, so the multiplication by m -scaler is closed.
01:04
So the sixth axiom is proved.
01:09
So based on this, we got this set.
01:10
We got these two operations.
01:12
We need to check if this, we need to verify if it is a vector space.
01:21
So let's start.
01:22
The first action was already proved.
01:23
It's just the closeness of this summation operation.
01:26
And the sixth axiom was also proved.
01:29
Is just the closed of the square multiplication.
01:34
So in this case, we need to prove the commutativity of the sum operation.
01:41
So u plus b, this is just a10 -0b1 plus a -2.
01:53
And this result we already know what it is from the previous slide.
01:58
So it's just a1, a2, 0, b1 plus b2.
02:05
Great.
02:06
And the right hand side of this equation is v plus u.
02:13
That means a20 -0b2 plus a1 -1.
02:20
A1 -0 -0 -b1.
02:23
And what happened here is that we obtain a2 plus a1 -0 -0 -b2 plus b1.
02:31
But these elements here are just real numbers, so they commute.
02:37
And this becomes a1 plus a2 -0 -b1 plus b -2.
02:45
That means that these both expressions are the same, so the second action is satisfied.
02:54
The next, here we need to define an extra vector that is the vector w, is a3b3, still diagonal matrix on m.
03:03
And we need to prove the associativity property.
03:08
That means u plus v plus w, that's going to be a10b1 plus here.
03:22
And the sum of these two matrices will be a2 plus a3 -0 -b2 plus b -3.
03:34
And then we need to sum these two matrices.
03:37
It here.
03:39
And what you obtain is a1 plus a2 plus a3 0, b1 plus b2 plus b3.
03:54
And then the right -hand side of this expression reads as a1 plus a2 0, b1 plus b2 plus a3 and this sum is equal to a1 plus a2 plus a3 0, b1 plus b2 plus b3.
04:35
And these two matrices are the same, so the third axiom is satisfied on this space.
04:45
Next, we need to define the zero vector, that means the ones that satisfy this property, which pick some arbitrary vector on the space and we sum with zero vector, we obtain the same vector.
04:58
And this part is just an addition of the commutativity property of the sum operation, but it is enough to prove just one part of it.
05:07
That means u plus the zero vector returns the same vector.
05:12
So the zero vector in this case is defined as the zero matrix.
05:21
And you can observe that it's still diagonal matrix.
05:23
It's not diagonal in the strict sense, but it satisfies the property that is required.
05:30
So this is part of this set.
05:34
And then we pick u plus the zero vector, in this case, is a100b1 plus 00...